© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L11n5 L11n7

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n6 Visit L11n6's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 X5,11,6,10 X11,19,12,18 X13,20,14,21 X19,5,20,22 X21,12,22,13 X2,16,3,15

Gauss Code:

{{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 10, -8, 4, 11, -2, 3, 7, -9, 8, -10, 9}}

Jones Polynomial:

- q-9/2 + 2q-7/2 - 4q-5/2 + 6q-3/2 - 7q-1/2 + 6q1/2 - 6q3/2 + 4q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-16 + 2q-14 - 3q-6 + 2 + 3q2 + q4 + 3q6 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - 2a-1z-1 - 4a-1z - 3a-1z3 - a-1z5 + 4az-1 + 6az + 3az3 - 3a3z-1 - 3a3z + a5z-1

Kauffman Polynomial:

- a-4z2 + 3a-4z4 - a-4z6 + 3a-3z - 9a-3z3 + 11a-3z5 - 3a-3z7 - a-2 + a-2z2 - 6a-2z4 + 10a-2z6 - 3a-2z8 - 2a-1z-1 + 13a-1z - 28a-1z3 + 20a-1z5 - 2a-1z7 - a-1z9 - 2 + 11z2 - 25z4 + 20z6 - 5z8 - 4az-1 + 20az - 30az3 + 13az5 - az9 - 3a2 + 12a2z2 - 18a2z4 + 9a2z6 - 2a2z8 - 3a3z-1 + 12a3z - 12a3z3 + 4a3z5 - a3z7 - a4 + 3a4z2 - 2a4z4 - a5z-1 + 2a5z - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               2 1   j = 4             4 2     j = 2         1 3 2       j = 0         5 4         j = -2       3 4           j = -4     1 3             j = -6   1 3               j = -8   1                 j = -10 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 6]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 6]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 14, 10, 15], > X[8, 4, 9, 3], X[5, 11, 6, 10], X[11, 19, 12, 18], X[13, 20, 14, 21], > X[19, 5, 20, 22], X[21, 12, 22, 13], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 10, -8, 4, 11, -2, 3, 7, > -9, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 2 4 6 7 3/2 5/2 -q + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 6 q + 4 q - 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 3 2 4 6 12 14 2 + q + --- - -- + 3 q + q + 3 q + q - q 14 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 6]][a, z]
Out[8]=	3 5 3 3 5 -2 4 a 3 a a z 4 z 3 z 3 z 3 z --- + --- - ---- + -- + -- - --- + 6 a z - 3 a z + -- - ---- + 3 a z - -- a z z z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 6]][a, z]
Out[9]=	3 5 -2 2 4 2 4 a 3 a a 3 z 13 z 3 -2 - a - 3 a - a - --- - --- - ---- - -- + --- + ---- + 20 a z + 12 a z + a z z z z 3 a a 2 2 3 3 5 2 z z 2 2 4 2 9 z 28 z 3 > 2 a z + 11 z - -- + -- + 12 a z + 3 a z - ---- - ----- - 30 a z - 4 2 3 a a a a 4 4 5 3 3 5 3 4 3 z 6 z 2 4 4 4 11 z > 12 a z - a z - 25 z + ---- - ---- - 18 a z - 2 a z + ----- + 4 2 3 a a a 5 6 6 7 7 20 z 5 3 5 6 z 10 z 2 6 3 z 2 z > ----- + 13 a z + 4 a z + 20 z - -- + ----- + 9 a z - ---- - ---- - a 4 2 3 a a a a 8 9 3 7 8 3 z 2 8 z 9 > a z - 5 z - ---- - 2 a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 2 1 1 1 3 1 3 3 5 + -- + q + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 4 t + 2 10 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 3 q t + 2 q t + 4 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n6

L11n5 L11n7